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Need help solving few questions

by sanalnnair » Tue Sep 07, 2010 8:01 pm
Hi, Can someone help solve the below questions and tell me the steps by which you got the answer. I am little confused and not able to get the correct answer.


1) If a committee of 3 people is to be selected from among 5 married couples so that the
committee does not include two people who are married to each other, how many such
committees are possible?

The answer given is 80.


2) The infinite sequence a1, a2,..., an,... is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

The answer given is 74.


3) In the xy-plane, the point (-2, -3) is the center of a circle. The point (-2, 1) lies inside the circle and the point (4, -3) lies outside the circle. If the radius r of the circle is an integer, then r =

The answer given is 5.


4) The figure above represents a square garden that is divided into 9 rectangular regions
with indicated dimensions in meters. The shaded regions are planted with peas, and the
unshaded regions are planted with tomatoes. If the sum of the areas of the regions
planted with peas is equal to the sum of the areas of the regions planted with tomatoes,
what is the value of x?


The answer given is 2.


5) What is the value of │x + 7│?
(1) │x + 3│= 14
(2) (x + 2)~2 = 169

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

The answer given is D.



Regards
[/img]
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by sanalnnair » Tue Sep 07, 2010 8:04 pm
I have attached Question 5 below in a word file as the characters were not showing up clearly.
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by sanju09 » Tue Sep 07, 2010 8:33 pm
sanalnnair wrote:Hi, Can someone help solve the below questions and tell me the steps by which you got the answer. I am little confused and not able to get the correct answer.


1) If a committee of 3 people is to be selected from among 5 married couples so that the
committee does not include two people who are married to each other, how many such
committees are possible?

The answer given is 80.


2) The infinite sequence a1, a2,..., an,... is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

The answer given is 74.


3) In the xy-plane, the point (-2, -3) is the center of a circle. The point (-2, 1) lies inside the circle and the point (4, -3) lies outside the circle. If the radius r of the circle is an integer, then r =

The answer given is 5.


4) The figure above represents a square garden that is divided into 9 rectangular regions
with indicated dimensions in meters. The shaded regions are planted with peas, and the
unshaded regions are planted with tomatoes. If the sum of the areas of the regions
planted with peas is equal to the sum of the areas of the regions planted with tomatoes,
what is the value of x?


The answer given is 2.


5) What is the value of │x + 7│?
(1) │x + 3│= 14
(2) (x + 2)~2 = 169

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

The answer given is D.



Regards
[/img]

Please post one question per thread, and also consider revealing its source. I will take them one by one only, starting from your first problem:

1) If a committee of 3 people is to be selected from among 5 married couples so that the
committee does not include two people who are married to each other, how many such
committees are possible?

The answer given is 80.

First person can be selected in 10 ways, second person can be selected in 8 ways, and the third person can be selected in 6 ways, in order to avoid spouseship between any selected two. Now, (10 × 8 × 6) gives us the various arrangements of those selected three, divide it by 3! to get the number of possible committees. 3! = 6, the right answer is therefore,[spoiler]80[/spoiler].
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by sanju09 » Tue Sep 07, 2010 8:53 pm
Your second question...

2) The infinite sequence a1, a2,..., an,... is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

The answer given is 74.

Most important is to note an = an-4 for n > 4 here. With this we can understand that a5 = a1 = 2, a6 = a2 = -3, a7 = a3 = 5, a8 = a4 = -1, and we can figure out 24 cycles of (2 - 3 + 5 - 1) uptil a96, and a97 = 2 again; the sum of the first 97 terms of the sequence

= 24 × (2 - 3 + 5 - 1) + 2

= 24 × 3 + 2

= [spoiler]74[/spoiler]
The mind is everything. What you think you become. -Lord Buddha



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by sanju09 » Tue Sep 07, 2010 9:08 pm
your third question...


3) In the xy-plane, the point (-2, -3) is the center of a circle. The point (-2, 1) lies inside the circle and the point (4, -3) lies outside the circle. If the radius r of the circle is an integer, then r =

The answer given is 5.

√[{-2 - (-2)} ^2 + (-3 - 1) ^2] < r < √[(-2 - 4) ^2 + {-3 - (-3) ^2}]

4 < r < 6, and r is an integer, hence r = [spoiler]5[/spoiler].
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by RCV » Tue Sep 07, 2010 9:39 pm
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by sanju09 » Tue Sep 07, 2010 9:46 pm
Your fourth question...

4) The figure above represents a square garden that is divided into 9 rectangular regions
with indicated dimensions in meters. The shaded regions are planted with peas, and the
unshaded regions are planted with tomatoes. If the sum of the areas of the regions
planted with peas is equal to the sum of the areas of the regions planted with tomatoes,
what is the value of x?


The answer given is 2.

Given is a square garden of side 9, first column from left includes 3 x + 3 x as shaded and 3 x as unshaded, second or the middle column includes 3 (6 - x) as shaded and 3 (6 - x) + 3 (6 - x) as unshaded, the rightmost column includes 3 × 3 + 3 × 3 as shaded and 3 × 3 as unshaded.

Since, the sum of the areas of the regions planted with peas is equal to the sum of the areas of the regions planted with tomatoes

Therefore, shaded area = unshaded area

Or, 3 x + 3 x + 3 (6 - x) + 3 × 3 + 3 × 3 = 3 x + 3 (6 - x) + 3 (6 - x) + 3 × 3

Or, 6 x + 18 - 3 x + 18 = 3 x + 36 - 6 x + 9

Or, 6 x = 9

My x is coming out to be [spoiler]1.5[/spoiler], not [spoiler]2[/spoiler].

Please check it again
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by sanju09 » Tue Sep 07, 2010 10:01 pm
And here comes your fifth and final question, I had to edit it twice just as to make sure that the modulus symbol is not creating bizarre results over and over again...

5) What is the value of ×€ x + 7×€ ?
(1) ×€ x + 3×€ is 14
(2) (x + 2) ^2 = 169

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is
sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

The answer given is D.


Regards

×€ x + 7×€ =?

(1) If ×€ x + 3×€ is 14, then either x + 3 = 14 gives x = 11 and ×€ x + 7×€ is 18 or x + 3 = -14 gives x = -17 and ×€ x + 7×€ is 10; two answers because x is either 11 or -17, insufficient

(2) I guess this reads (x + 2) ^2 = 169. Again, x + 2 = ±13 would lead to two answers because x is either 11 or -15, insufficient

When taken together, x is clearly 11 and ×€ x + 7×€ is 18.


[spoiler]C

Who said D[/spoiler]? please check it again
Last edited by sanju09 on Tue Sep 07, 2010 10:47 pm, edited 2 times in total.
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by inactived » Tue Sep 07, 2010 10:03 pm
Hi sanalnnair,

Can you repost question #5? I can't understand what the symple ~ stands for?

Tks.
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by sanalnnair » Tue Sep 07, 2010 10:21 pm
I have posted the question in a word attachment in the start of my forum. Please go thru the attachment
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by sanju09 » Tue Sep 07, 2010 10:26 pm
inactived wrote:Hi sanalnnair,

Can you repost question #5? I can't understand what the symple ~ stands for?

Tks.
I think it's meant for raising to some power of (x + 2).

The modulus symbol in fifth question is giving some bizarre results, I tried to edit it once with a different font for the modulus symbol "×€x×€" , which is lucid here, but few issues are still existing with this too. I must edit it again, posters can copy paste this symbol, since it is lucid here, at the least.
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by sanalnnair » Wed Sep 08, 2010 9:01 am
Hi Sanju009,


I had read the answers of those questions which I had posted. Many thanks for that.


I just have a single doubt in this question.

Wed Sep 08, 2010 9:53 am
Your second question...

2) The infinite sequence a1, a2,..., an,... is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

The answer given is 74.

Most important is to note an = an-4 for n > 4 here. With this we can understand that a5 = a1 = 2, a6 = a2 = -3, a7 = a3 = 5, a8 = a4 = -1, and we can figure out 24 cycles of (2 - 3 + 5 - 1) uptil a96, and a97 = 2 again; the sum of the first 97 terms of the sequence

= 24 X (2 - 3 + 5 - 1) + 2

= 24 X 3 + 2

= 74


Everything is clear, but why did you add 2 with 72. I understood how you got 72, but since there are 24 cycles of those 4 numbers, what is the need or why would you add 2 to it??

Please help
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by sanju09 » Wed Sep 08, 2010 8:10 pm
sanalnnair wrote:Hi Sanju009,


I had read the answers of those questions which I had posted. Many thanks for that.


I just have a single doubt in this question#2.


Everything is clear, but why did you add 2 with 72. I understood how you got 72, but since there are 24 cycles of those 4 numbers, what is the need or why would you add 2 to it??

Please help
Hi sanalnnair,

First promise three things (1) you would never add an extra '0' to my name, I'm no JamesBond007 (2) you would always post only one question per thread (3) you'll consider quoting what's not yours in your posts, DONE?

24 cycles of those 4 numbers include first 96 terms of the sequence, cycle restarts at a97, which is got to be 2 again. They want us get the sum of the first 97 terms of the sequence, 72 is the sum of the first 96 terms of the sequence, add 2 to it and get the sum of the first 97 terms of the sequence as 74.
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by sanalnnair » Thu Sep 09, 2010 9:02 am
Hi Sanju09. I am sorry. Its just that when I was replying in hurry, I had put an extra 0. You can call me by my name Sanal.


I agree on all of the 3 conditions which you have told me. I will post 1 question per post. and what did you mean by posting quotes??

Is it like how you post a specific quote "It takes less time to do something right than to explain why you did it wrong."??


I don't have my own quote, but I will try to find a good one for me.. :)
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by sanju09 » Fri Sep 10, 2010 12:46 am
sanalnnair wrote:Hi Sanju09. I am sorry. Its just that when I was replying in hurry, I had put an extra 0. You can call me by my name Sanal.


I agree on all of the 3 conditions which you have told me. I will post 1 question per post. and what did you mean by posting quotes??

Is it like how you post a specific quote "It takes less time to do something right than to explain why you did it wrong."??


I don't have my own quote, but I will try to find a good one for me.. :)
No, with quoting I never meant that. Please see your previous post in this thread where you've copied my material obviously with an intent of reference, but if you do not quote that as under, then it could amount to a crime called PLAGIARISM in third world:
Your second question...

2) The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?

The answer given is 74.

Most important is to note an = an-4 for n > 4 here. With this we can understand that a5 = a1 = 2, a6 = a2 = -3, a7 = a3 = 5, a8 = a4 = -1, and we can figure out 24 cycles of (2 - 3 + 5 - 1) uptil a96, and a97 = 2 again; the sum of the first 97 terms of the sequence

= 24 X (2 - 3 + 5 - 1) + 2

= 24 X 3 + 2

= [spoiler]74[/spoiler]
:|
The mind is everything. What you think you become. -Lord Buddha



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